Can I substitute Carrot Juice for Tomato Soup, in 'Tomato Soup cake'? Do I need to adjust the recipe in other ways?
Tomato soup is thicker and richer than carrot juice. Using carrot juice will change the moisture content of the cake, and it will not rise or bake properly.
Grab some vegetable juice (V8 or something like it) and bring it to a gentle simmer. Put a teaspoon of corn starch in 3/4 tablespoon of water, stir the mixture rapidly for a bit, then stir that rapidly into the vegetable juice. Taste for salt. Bring it briefly to a rolling boil and then take it off the heat. It should coat the back of a spoon.
Keep stirring, occasionally as it cools, and you'll have something you can use. You can't directly substitute carrot juice for condensed soup, and simply thickening carrot juice in a similar manner as what I described above is going to give the cake strange sweet and wet spots after it bakes. I share Sharlyn's concern about the cake rising properly and evenly.
You could try a puree of carrot, thicken it a bit as described above, and go with that, but the minimal viscosity you need is probably that of vegetable juice after being thickened.
Adding a similar amount of juice as what is contained in a can of soup (10-11 ounces) might be too much liquid since the soup is condensed (tomato paste, corn syrup, flour). I haven't tried it, but it might work if you used 1/2 cup of juice, and increased the flour by 1/4 cup.