# Vapor Comression Refrigeration cycle

I am doing a project for my thermodynamics class and we have to choose a compressor for a a vapor compression refrigeration cycle that has a heat transfer rate of 1500 btu/h and must maintain 0F in the freezer when the room temp is 90F. I have went through some calculations to find the mass flow rate and compressor power but i have to estimate the length of compressor tubing. I am using the eqn Q=hA(Tfluid-Tsurrouding) I chose to use copper as the tubing but I cannot find h or the temperature of the fluid. Any help with this is appreciated. I also need help with what the tube inside and outside diameter should be.

Also what temperatures should the cycle be designed for based on the contraints given?

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the fluid is r-134a also

Mech_Engineer
Gold Member
Sounds like you'll have to calculate the free convection coefficient on the outside of the tube, as well as the forced convection coefficient on the inside of the tube. Have you taken a heat trasfer class yet?

Sounds like you'll have to calculate the free convection coefficient on the outside of the tube, as well as the forced convection coefficient on the inside of the tube. Have you taken a heat trasfer class yet?
heat transfer is next semester. The problem can get very indepth but the design is supposed to be simple. I just need to know what temperatures i should design it for and how to calculate the h for that above equation. I can find the mass flow rate from the eqn
Qin=m(h1-h4) since Qin is given but i got a flow rate of .0034 kg/s which seems very low using temperatures of 100.4F and -13F

Mech_Engineer
Gold Member
If you're given the power of the system (1500BTU/hr = ~440 W) and the temperature of the hot and cold regions being used by the refrigeration system, it sounds like you have everything you need. Do you have to take any efficiencies into account? Are any efficiencies given to you for the heat exchangers or turbine/expansion valve?

What assumptions have you made to fix the states of the gas at states 1 & 4? Saturated vapor? Your application of the equation that determines the mass flow across the heat exchanger is correct, as long as you have properly fixed states 1 & 4 and looked up the correct enthalpy values.

But i heard from a few ppl that you have to choose temperatures above the hot region and below the cold region i believe due to efficiencies since no cycle is 100% efficient. The compressor power can be found from the eqn W=m(h2-h1) I took the 1500btu/hr and converted that to .4396 kj/s and found the mass flow rate.

State 1- sat vapor T= -25C i interpolated for h1, p1 and s1

State 2 s1=s2 T = 38C . i interpoated in superheated tables to get h2

state 3 sat liq 38C

State 4 h3=h4

i know how to go about the process i just need to double check on what temperatures to use and how to find the length of the condeser tube and size of the tube. I can use Qout=hA(/\T) to solve for L which comes from A=2piL but i cannot also find h. And the delta T is (Tfluid-Tsurroundings). How do I find the T fluid?

Mech_Engineer
Gold Member
But i heard from a few ppl that you have to choose temperatures above the hot region and below the cold region i believe due to efficiencies since no cycle is 100% efficient.
That's true, but you can't really choose any old arbitrary number. I suppose if you haven't taken heat transfer yet, this is the only way to do it though...

Are you supposed to calculate the convective coefficient (h) around the pipe even though you've never taken heat transfer, or are you given something to use?

we arent given anything else to use. the problem says Select a matierial and tube size for the condenser tubing an estimate the total condesor tube length. I heard from a couple ppl in class about that equation with h in it and you can solve for L. But from what i understand you have to calculate h it is not just an material property. And i have no clue what advantages there are to tube diameter being greater or smaller

The problem states:

A vapor compression regrigeration system using r134a is being designed for a household food freezer. The refrigeration system must maintain a temp. of 0F within the freezer compartment when the temp of the room is 90F. Under these conditions, the steady state heat transfer rate from the room into the freezer compartment is 1500btu/h. Specify operating pressures and temperatures at key points within the refrigeration system and estimate the refrigerant mass flow rate and compressor power required. Design for capacity factor of about 50%(i.e the compressor will run about 50% of the time). Select a compressor for the design from among the available sizes from suppliers. Select material and tube size for the condenser tubing, and estimate the total condenser tube length. Estimate the annual cost of operatin for the system.

Mech_Engineer
Gold Member
Well, zooming you ahead about a semester into the future, here's how to calculate the convective coefficient around a horizontal tube:

First, calculate the Rayleigh number (dimensionless parameter):

Rayleigh Number-
$$Ra_{D}=\frac{g\beta\left(T_{s}-T_{inf}\right)}{\nu\alpha}$$

Where g is gravity, $$\beta=1/T_{s}$$ for a perfect gas (T in kelvin), $$\nu$$ is the kinematic viscosity of air, and $$\alpha$$ is the thermal diffusivity of air.

Next, the Nusselt number (also dimensionless):

Nusselt number-
$$\overline{Nu}_{D}=\left\{0.60+\frac{0.387Ra_{D}^{1/6}}{\left[1+\left(0.559/Pr\right)^{9/16}\right]^{8/27}}\right\}^{2}$$

Where the Prandtl number (Pr) is a propety of air, Pr=0.697 at 367K

Finally, the convective coefficient is:
$$\overline{h}=\frac{k}{D}\overline{Nu}_{D}$$
Where k is the thermal conductivity of air

So, the heat loss per unit length of pipe is:
$$q'=\overline{h}*P*\left(T_{s}-T_{inf}\right)$$ (P is the perimeter of the pipe, pi times D)

Or heat loss over the length of the pipe is heat loss per unit length times the length of the pipe.

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does Pr vary depeding on what temperature it is at? if so is there a table to show these value

Mech_Engineer
Gold Member
does Pr vary depeding on what temperature it is at? if so is there a table to show these value
Pr does vary with temperature, I have a table in the back of my book. For 300K (approx 90 degrees F) $$Pr=0.707$$ for air. It is actually just a ratio of the kinematic viscosity and thermal diffusivity of air: http://en.wikipedia.org/wiki/Prandtl_number

Additionally, $$\nu=15.89*10^{-6}\frac{m^{2}}{s}$$, $$\alpha=22.5*10^{-6}\frac{m^{2}}{s}$$, and $$k=26.3*10^{-3}\frac{W}{m*K}$$ at 300K for air.

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would the eqn h=k/d work? i saw it in wikipedia

Mech_Engineer
Gold Member
would the eqn h=k/d work? i saw it in wikipedia
That sounds like a heat conduction equation to me, in which case no it will not work. Got a Wikipedia link?

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Mech_Engineer
Gold Member
In the above equations for calculating free convection, I discovered a slight typo in the calculation of the Rayleigh number. The equation was missing an L^3 term (L being the characteristic length; for a tube the characteristic length is the diameter). The corrected equation is:

$$Ra_{D}=\frac{g\beta\left(T_{s}-T_{inf}\right)D^{3}}{\nu\alpha}$$

Redbelly98
Staff Emeritus
Homework Helper
The problem states:
... Design for capacity factor of about 50%(i.e the compressor will run about 50% of the time)...
I'm sorry I can't help with the convection calculations, but don't you really want the compressor to handle 3000 btu/hr or 880 Watts?

Mech_Engineer
Gold Member
I'm sorry I can't help with the convection calculations, but don't you really want the compressor to handle 3000 btu/hr or 880 Watts?
Yup, if the compressor is meant to have a 50% duty cycle, it will have to have twice the minimum required capacity.

It looks to me like the condenser tube length will have to be very long (on the order of a couple hundred feet) and fairly hot (around 340K) if you only consider it a cylindrical tube with no fins attached. This is obviously why real freezers' condensers are cooled with large heatsink arrays that maximize surface area to minimize fluid path and temperature difference required.

Thank you my friend on this subject interesting and wonderful